In this survey article, we present an overview of a large class of probabilistic representations of the fragmentation equation, and we develop and study the interconnections in between these representations. We focus on the stochastic process which represents the evolution of the mass of a typical particle which undergoes fragmentation in time. These probabilistic representations range from Markov chains to stochastic differential equations with jumps, and we aim at constructing how they are inter-related. In particular, we show how these representations can be used to develop easy to implement numerical methods.
References
Ackleh, A. S., Lyons, R. and Saintier, N. (2021). A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models. ESAIM Math. Model. Numer. Anal. 55 2473–2501. MR4330732Ackleh, A. S., Lyons, R. and Saintier, N. (2021). A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models. ESAIM Math. Model. Numer. Anal. 55 2473–2501. MR4330732
Ackleh, A. S. and Saintier, N. (2020). Well-posedness of a system of transport and diffusion equations in space of measures. J. Math. Anal. Appl. 492 124397, 28. MR4126768Ackleh, A. S. and Saintier, N. (2020). Well-posedness of a system of transport and diffusion equations in space of measures. J. Math. Anal. Appl. 492 124397, 28. MR4126768
Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48. MR1673235Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48. MR1673235
Azaïs, R. (2014). A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process. ESAIM Probab. Stat. 18 726–749. MR3334012Azaïs, R. (2014). A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process. ESAIM Probab. Stat. 18 726–749. MR3334012
Azaïs, R. and Muller-Gueudin, A. (2016). Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes. Electron. J. Stat. 10 3648–3692. MR3579198Azaïs, R. and Muller-Gueudin, A. (2016). Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes. Electron. J. Stat. 10 3648–3692. MR3579198
Bailleul, I. F., Man, P. L. W. and Kraft, M. (2010). A stochastic algorithm for parametric sensitivity in Smoluchowski’s coagulation equation. SIAM J. Numer. Anal. 48 1064–1086. MR2679572Bailleul, I. F., Man, P. L. W. and Kraft, M. (2010). A stochastic algorithm for parametric sensitivity in Smoluchowski’s coagulation equation. SIAM J. Numer. Anal. 48 1064–1086. MR2679572
Banasiak, J. and Arlotti, L. (2006). Perturbations of Positive Semigroups with Applications. Springer Monographs in Mathematics. Springer London, Ltd., London. MR2178970Banasiak, J. and Arlotti, L. (2006). Perturbations of Positive Semigroups with Applications. Springer Monographs in Mathematics. Springer London, Ltd., London. MR2178970
Berestycki, J., Bertoin, J., Haas, B. and Miermont, G. (2010). Quelques aspects fractals des fragmentations aléatoires. In Quelques Interactions Entre Analyse, Probabilités et Fractals. Panor. Synthèses 32 191–243. Soc. Math. France, Paris. MR2932439Berestycki, J., Bertoin, J., Haas, B. and Miermont, G. (2010). Quelques aspects fractals des fragmentations aléatoires. In Quelques Interactions Entre Analyse, Probabilités et Fractals. Panor. Synthèses 32 191–243. Soc. Math. France, Paris. MR2932439
Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge. MR2253162Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge. MR2253162
Bertoin, J. (2019). On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors. J. Funct. Anal. 277 108270, 29. MR4013826Bertoin, J. (2019). On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors. J. Funct. Anal. 277 108270, 29. MR4013826
Beznea, L., Deaconu, M. and Lupaşcu, O. (2015). Branching processes for the fragmentation equation. Stochastic Process. Appl. 125 1861–1885. MR3315615Beznea, L., Deaconu, M. and Lupaşcu, O. (2015). Branching processes for the fragmentation equation. Stochastic Process. Appl. 125 1861–1885. MR3315615
Beznea, L., Deaconu, M. and Lupaşcu, O. (2016). Stochastic equation of fragmentation and branching processes related to avalanches. J. Stat. Phys. 162 824–841. MR3456978Beznea, L., Deaconu, M. and Lupaşcu, O. (2016). Stochastic equation of fragmentation and branching processes related to avalanches. J. Stat. Phys. 162 824–841. MR3456978
Beznea, L., Deaconu, M. and Lupaşcu-Stamate, O. (2019). Numerical approach for stochastic differential equations of fragmentation; application to avalanches. Math. Comput. Simulation 160 111–125. MR3926782Beznea, L., Deaconu, M. and Lupaşcu-Stamate, O. (2019). Numerical approach for stochastic differential equations of fragmentation; application to avalanches. Math. Comput. Simulation 160 111–125. MR3926782
Bouguet, F. (2018). A probabilistic look at conservative growth-fragmentation equations. In Séminaire de Probabilités XLIX. Lecture Notes in Math. 2215 57–74. Springer, Cham. MR3837101Bouguet, F. (2018). A probabilistic look at conservative growth-fragmentation equations. In Séminaire de Probabilités XLIX. Lecture Notes in Math. 2215 57–74. Springer, Cham. MR3837101
Bouteiller, C. L., Naaim-Bouvet, F., Mathys, N. and Lavé, J. (2011). A new framework for modeling sediment fining during transport with fragmentation and abrasion. Journal of Geophysical Research 116. 10.1029/2010jf001926Bouteiller, C. L., Naaim-Bouvet, F., Mathys, N. and Lavé, J. (2011). A new framework for modeling sediment fining during transport with fragmentation and abrasion. Journal of Geophysical Research 116. 10.1029/2010jf001926
Cañizo, J. A., Gabriel, P. and Yoldaş, H. (2021). Spectral gap for the growth-fragmentation equation via Harris’s theorem. SIAM J. Math. Anal. 53 5185–5214. MR4312814Cañizo, J. A., Gabriel, P. and Yoldaş, H. (2021). Spectral gap for the growth-fragmentation equation via Harris’s theorem. SIAM J. Math. Anal. 53 5185–5214. MR4312814
Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285. MR4268042Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285. MR4268042
Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285. MR4268042Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285. MR4268042
Çinlar, E. and Jacod, J. (1981). Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981). Progr. Prob. Statist. 1 159–242. Birkhäuser, Boston, Mass. MR0647786Çinlar, E. and Jacod, J. (1981). Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981). Progr. Prob. Statist. 1 159–242. Birkhäuser, Boston, Mass. MR0647786
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I, 2nd ed. Probability and Its Applications (New York). Springer, New York. Elementary theory and methods. MR1950431Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I, 2nd ed. Probability and Its Applications (New York). Springer, New York. Elementary theory and methods. MR1950431
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II, 2nd ed. Probability and Its Applications (New York). Springer, New York. General theory and structure. MR2371524Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II, 2nd ed. Probability and Its Applications (New York). Springer, New York. General theory and structure. MR2371524
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B 46 353–388. With discussion. MR0790622Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B 46 353–388. With discussion. MR0790622
Diemer, R. B. and Olson, J. H. (2002). A moment methodology for coagulation and breakage problems: Part 3—generalized daughter distribution functions. Chemical Engineering Science 57 4187–4198. 10.1016/S0009-2509(02)00366-4Diemer, R. B. and Olson, J. H. (2002). A moment methodology for coagulation and breakage problems: Part 3—generalized daughter distribution functions. Chemical Engineering Science 57 4187–4198. 10.1016/S0009-2509(02)00366-4
Doumic, M., Escobedo, M. and Tournus, M. (2018). Estimating the division rate and kernel in the fragmentation equation. Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 1847–1884. MR3906858Doumic, M., Escobedo, M. and Tournus, M. (2018). Estimating the division rate and kernel in the fragmentation equation. Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 1847–1884. MR3906858
Dubovski˘ı, P. B. and Stewart, I. W. (1996). Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci. 19 571–591. MR1385155Dubovski˘ı, P. B. and Stewart, I. W. (1996). Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci. 19 571–591. MR1385155
Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge. Revised reprint of the 1989 original. MR1932358Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge. Revised reprint of the 1989 original. MR1932358
El Karoui, N. and Lepeltier, J.-P. (1977). Représentation des processus ponctuels multivariés à l’aide d’un processus de Poisson. Z. Wahrsch. Verw. Gebiete 39 111–133. MR0448546El Karoui, N. and Lepeltier, J.-P. (1977). Représentation des processus ponctuels multivariés à l’aide d’un processus de Poisson. Z. Wahrsch. Verw. Gebiete 39 111–133. MR0448546
Engel, K.-J. and Nagel, R. (2000). One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, New York. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR1721989Engel, K.-J. and Nagel, R. (2000). One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, New York. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR1721989
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. Characterization and convergence. MR0838085Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. Characterization and convergence. MR0838085
Feinberg, E. A., Mandava, M. and Shiryaev, A. N. (2014). On solutions of Kolmogorov’s equations for nonhomogeneous jump Markov processes. J. Math. Anal. Appl. 411 261–270. MR3118483Feinberg, E. A., Mandava, M. and Shiryaev, A. N. (2014). On solutions of Kolmogorov’s equations for nonhomogeneous jump Markov processes. J. Math. Anal. Appl. 411 261–270. MR3118483
Feller, W. (1945). Erratum to “On the integro-differential equations of purely discontinuous Markoff processes”. Trans. Amer. Math. Soc. 58 474–474. 10.1090/S0002-9947-45-99962-4Feller, W. (1945). Erratum to “On the integro-differential equations of purely discontinuous Markoff processes”. Trans. Amer. Math. Soc. 58 474–474. 10.1090/S0002-9947-45-99962-4
Hoang, V. H., Pham Ngoc, T. M., Rivoirard, V. and Tran, V. C. (2022). Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation. Scand. J. Stat. 49 4–43. MR4391046Hoang, V. H., Pham Ngoc, T. M., Rivoirard, V. and Tran, V. C. (2022). Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation. Scand. J. Stat. 49 4–43. MR4391046
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam-New York; Kodansha, Ltd., Tokyo. MR0637061Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam-New York; Kodansha, Ltd., Tokyo. MR0637061
Jacod, J. (1974/75). Multivariate point processes: Predictable projection, Radon-Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253. MR0380978Jacod, J. (1974/75). Multivariate point processes: Predictable projection, Radon-Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253. MR0380978
Jacod, J. (1977). Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales. In Séminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976). Lecture Notes in Math., Vol. 581 390–410. Springer, Berlin. MR0458578Jacod, J. (1977). Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales. In Séminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976). Lecture Notes in Math., Vol. 581 390–410. Springer, Berlin. MR0458578
Jourdain, B. (2004). Uniqueness via probabilistic interpretation for the discrete coagulation fragmentation equation. Commun. Math. Sci. 2 75–83. MR2119875Jourdain, B. (2004). Uniqueness via probabilistic interpretation for the discrete coagulation fragmentation equation. Commun. Math. Sci. 2 75–83. MR2119875
Kallenberg, O. (2021). Foundations of Modern Probability. Probability Theory and Stochastic Modelling 99. Springer, Cham. Third edition [of 1464694]. MR4226142Kallenberg, O. (2021). Foundations of Modern Probability. Probability Theory and Stochastic Modelling 99. Springer, Cham. Third edition [of 1464694]. MR4226142
Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications. MR1207584Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications. MR1207584
Kolmogoroff, A. N. (1941). Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 99–101. MR0004415Kolmogoroff, A. N. (1941). Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 99–101. MR0004415
Kolokoltsov, V. N. (2006). Kinetic equations for the pure jump models of k-nary interacting particle systems. Markov Process. Related Fields 12 95–138. MR2223422Kolokoltsov, V. N. (2006). Kinetic equations for the pure jump models of k-nary interacting particle systems. Markov Process. Related Fields 12 95–138. MR2223422
Lamb, W. (2004). Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Methods Appl. Sci. 27 703–721. MR2070223Lamb, W. (2004). Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Methods Appl. Sci. 27 703–721. MR2070223
Lemaire, V., Thieullen, M. and Thomas, N. (2020). Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris-Lecar model. Adv. in Appl. Probab. 52 138–172. MR4092810Lemaire, V., Thieullen, M. and Thomas, N. (2020). Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris-Lecar model. Adv. in Appl. Probab. 52 138–172. MR4092810
Lewis, P. A. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26 403–413. MR0546120Lewis, P. A. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26 403–413. MR0546120
Man, P. L. W., Norris, J. R., Bailleul, I. F. and Kraft, M. (2010). Coupling algorithms for calculating sensitivities of Smoluchowski’s coagulation equation. SIAM J. Sci. Comput. 32 635–655. MR2609334Man, P. L. W., Norris, J. R., Bailleul, I. F. and Kraft, M. (2010). Coupling algorithms for calculating sensitivities of Smoluchowski’s coagulation equation. SIAM J. Sci. Comput. 32 635–655. MR2609334
Martinez-Bazán, C., Rodriguez-Rodriguez, J., Deane, G. B., Montañes, J. L. and Lasheras, J. C. (2010). Considerations on bubble fragmentation models. Journal of Fluid Mechanics 661 159–177. 10.1017/s0022112010003186Martinez-Bazán, C., Rodriguez-Rodriguez, J., Deane, G. B., Montañes, J. L. and Lasheras, J. C. (2010). Considerations on bubble fragmentation models. Journal of Fluid Mechanics 661 159–177. 10.1017/s0022112010003186
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. Itô calculus, Reprint of the second (1994) edition. MR1780932Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. Itô calculus, Reprint of the second (1994) edition. MR1780932
Stewart, I. W. (1989). A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11 627–648. MR1011810Stewart, I. W. (1989). A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11 627–648. MR1011810
Tyran-Kamińska, M. (2009). Substochastic semigroups and densities of piecewise deterministic Markov processes. J. Math. Anal. Appl. 357 385–402. MR2557653Tyran-Kamińska, M. (2009). Substochastic semigroups and densities of piecewise deterministic Markov processes. J. Math. Anal. Appl. 357 385–402. MR2557653